Online Activity for Slope and Slope-Intercept Linear Equations

by: Algebra Funsheets - November 22nd, 2008   2 Comments

roachlinearEqns
The other day, my students spent the day in the computer lab and explored the myriad of web sites on line that let them explore linear equations. I directed them to some fairly decent tutorials, and various online tools that allowed them to physically move lines to explore the effect it had on slope and equations. It was the first time I took them to the lab this year, and thought they would thoroughly enjoy this hands on opportunity. But they blandly went through the tutorials, and some even complained that they would prefer next time to stay in the class room, because it’s more interesting when I explain things to them.

However, the last 15-20 minutes of the day, I directed them to the website below. The student loved it. Many students who struggled with rate of change/slope and writing slope-intercept equations were proficient by the end of class. Unlike most other math “games,” the graphics on this are excellent and students actually enjoy the concept of the game, which is rather strange (which is why they liked it). There are roaches that traverse the coordinate plane at a particular line, for which the students must identify the slope or slope-intercept equation that will exterminate the roaches. They can even choose the method of exterminating the roaches (fumigator, shoe …).

Linear equations game

If you don’t have the time to do this during a class period, show the students the site and they will go there on their own time. Many of my students asked me to send them the link at their home address so they could play at home!

Slide and Divide Method of Factoring Trinomials

by: Algebra Funsheets - November 22nd, 2008   18 Comments

FactoringTri
Someone asked about how to teach factoring trinomials where the leading coefficient is not 1, so I thought I would post it here as well.

The best method I’ve found is often referred to as the slide and divide method, for trinomials of the type ax^2 + bx + c

Anyways, take the trinomial 3x^2 + x – 10.
(Always ensure to pull out any common factors before continuing with this method).
First you slide the leading coefficient to the end and multiply it by the constant:
You get x^2 + x – 30
Then factor like normal
(x + 6)(x – 5)
You have to “put back” the number you slid out, so to speak. You do this by dividing the constant in each factor by the leading coefficient you “slid” out of the way.
(x + 6/3)(x – 5/3)
Simplify the fractional terms you end up with.
(x + 2)(x – 5/3)
Once it’s simplified, if there’s a fraction left, the denominator becomes the coefficient of the variable term.
ANSWER: (x + 2)(3x – 5)

Here’s another example without the explanations:

2x^2 – 7x + 5
x^2 – 7x + 10
(x – 5)(x – 2)
(x – 5/2)(x – 2/2)
(x – 5/2)(x – 1)
ANSWER: (2x – 5)(x – 1)

Finally, an example where you have to pull out a common factor first:
12x^10 + 42x^9 + 18x^8
6x^8(2x^2 + 7x + 3)
6x^8(x^2 + 7x + 6)
6x^8(x + 6)(x + 1)
6x^8(x + 6/2)(x + 1/2)
6x^8(x + 3)(x + 1/2)
6x^8(x + 3)(2x + 1)

Teaching Slope of Horizontal and Vertical Lines.

by: Algebra Funsheets - November 19th, 2008   10 Comments

lines
I’ve arrived at that one frustrating point in teaching Algebra – teaching linear equations. I don’t know why the concepts are so difficult for students to grasp. I’ve tried teaching it a myriad of ways, and have yet to discover the best method for teaching this section. It takes me nearly a month to teach all the things that go with slope, slope-intercept and standard form equations. And those are just the basic concepts of linear equations. (After the tortuous month is over, something kicks in and they finally get it and claim it’s SO easy).

And why it is difficult for students to remember that horizontal lines have a slope of zero and vertical lines have undefined slopes is beyond me.

I always start by trying to show them how to arrive at that conclusion using the slope formula. Since vertical lines invariably end up having zero in the denominator – which is THE big mathematical no-no – then their slope is undefined. They insist on calling it zero.

I resort to making them actually divide by zero on the calculator (after having arrived at some vertical line ratio using their slope formula), because it gives them “ERROR” since dividing by zero is a big mathematical no-no. And even though it says ERROR on their calculator, they still want to call it zero. The few who can actually remember that it’s undefined don’t want to call it undefined – they call it unfined, unrefined, or unidentified …

We analyze how linear equations with positive slopes increase or rise (from left to right) and how equations with negative slopes decrease or fall. Then we contrast that with a horizontal line, which neither rises or falls – and since it does “nothing” then it is “nothing” (0). But with a vertical line, you can’t tell if it is rising or falling; therefore since you can’t DEFINE which way the line is going, it is UNdefined.

Then I get desperate and resort to telling them that hOrizOntal has O’s in it, which looks like zeroes, because horizontal lines have a slope of zero. And that Vertical starts with a V, which strongly resembles the U in Undefined.

…that horiZontal has the letter Z in it, just like Zero …
…that Zero starts with a letter Z which actually has not one, but two horizontal lines ….

T-Chart Adding and Subtracting Integers

by: Algebra Funsheets - November 3rd, 2008   Add Comment

int2
I’ve found a method that the students have grasped fairly quickly for performing integer operations. It’s called the T-chart, and it works very well. The student creates a two column chart, one column is labeled with a + and the other with a -. Then the student places the values in their appropriate column. If the numbers are already in the same column, they add. If they are in different columns, they move the smaller of the two and subtract.

For example:
3 + -11

+ | -
3 | 11 (To represent 3 positives and 11 negatives)

They move the 3 under the 11 like so:
+ | -
__| 11
__| 3
(Ignore the underscores, this site won’t leave the space on the web page)
Since they moved the number, they subtract and get 8. Since 8 is in the negative column, the answer is -8.

Another example:
-3 + -9

+ | -
__| 3
__| 9
Since the two numbers are already in the same column, they add to get 12, and since it’s in the negative column, the answer is -12.

Hope this helps!

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